Three dimensional analysis of chromatic
aberration in diffractive elements with extended
depth of focus
D. Mas*, J. Espinosa, J. Perez, and C. Illueca
1
Dept. Optica, Univ. Alicante. Carretera San Vicente del Raspeig s/n - 03690 San Vicente del Raspeig Alicante
Spain
*Corresponding author: [email protected]
Abstract: The paper presents the polychromatic analysis of two diffractive
optical elements with extended depth of focus: the linear axicon and the
light sword optical element. Chromatic aberration produces axial
displacement of the focal segment line. Thus, we explore the possibility of
extending the focal depth of these elements to permit superposition of the
chromatic foci. In the case of an axicon, we achieve an achromatic zone
where focusing is produced. In the case of the light sword element, we show
that the focusing segment is out of axis. Therefore a superposition of colors
is produced, but not on axis overlapping. Instead, three colored and
separated foci are simultaneously obtained in a single plane. Three
dimensional structures of the propagated beams are analyzed in order to
provide better understanding of the properties and applications of such
elements.
© 2007 Optical Society of America
OCIS codes: (050.1970) Diffractive optics; (120.3620) Lens system design; (110.4100)
Modulation transfer function
References
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Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17842
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1. Introduction
Optical elements with extended depth of focus elements are of great interest that is increasing
in the last few years. Several configurations are proposed in the bibliography [1-10]. Some of
them are based on single elements [1,2,4,6-9,10] while others are based on convenient
association of several elements [3,5,8]. Extended depth of focus in a single element is usually
achieved by spatial multiplexing of phase information. We can find elements in which
focusing information is codified in radial [9,11] or angular coordinates [2] or even with
random localization through the surface of the element [7]. All these elements share the
common property that they produce a long and narrow focal segment which allows imaging in
a continuous region in the image domain. The effect can be achieved by multiplexing a
continuous range of focalization distances in the element, which can be done either with
refractive media [1,3,4,10] or by diffractive optical elements (DOEs) [2,6,7,9,12,13]. Among
these systems, we are most interested in single diffractive structures, which can be easily
implemented in a spatial light modulator (SLM). In particular we will focus our work on two
particular elements: Light Sword Optical Elements (LSOE) and Fresnel Axicons (FA) Optical
properties of these elements are widely known [1-3,11,15] and their performance has been
optimized in several ways [16-19].
Nevertheless, we have not found many studies about performance of these elements under
polychromatic illumination and none about chromatic aberrations in LSOEs. The major part
of published studies are devoted to compensate chromatic aberration [5,8] or to produce on
axis energy stabilization [16]. In our opinion, the special design of these elements together
with polychromatic illumination lead to interesting properties in the propagated field that, to
our knowledge, have not been already analyzed. Here we analyze these properties.
As it is widely known, chromatic aberration in DOEs produces an axial displacement of
the light patterns. Thus, if we illuminate an input element with three different wavelengths,
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we will obtain displaced colored replicas of the field distributions for each wavelength. The
DOEs we have just introduced are designed to provide extended focal depth. Thus they will
produce segment line whose extension is determined by the parameters of the DOE. In this
communication we study the performance of these elements under chromatic illumination.
We will use three different wavelengths, one per chromatic channel and we will obtain the
displaced focal segments. We will explore the possibility of extending this focal segment in
order to produce overlapping of the three segments. In the case of the axicon, we will show
that it is possible to obtain a focal segment where all colors overlap, thus producing the effect
of an achromatic axicon. In the case of a LSOE, this element produces an extra-axial segment
that rotates around the optical axis. Chromatic illumination will provide presence of the three
segments in the same region of the space but not axial overlapping segments. Thus, as we will
show later, it is possible to obtain a range of distances where three separated focus are
obtained simultaneously in the same plane. Obtaining a single element achromatic axicon
could lead to new applications in instrumental optics and ophthalmology, since alignment of
two compensating elements is not necessary. The LSOE in a trifocal chromatic configuration
may be also used in optical information processing since it allows focusing and chromatic
separation in one single element.
In this work we study chromatic aberration under different points of view. First, we will
analyze on axis irradiance. This common analysis is incomplete since it does not provide full
information about quality of the peak or the amount of light outside the optical axis.
Therefore, we also evaluate the quality of a peak by analyzing the width of the radial MTF at
each plane. Here we take profit of our previous design of fast algorithms [20-21] that compute
field distributions at any propagation plane. We compute the evolution of the MTF with the
propagated distance for three wavelengths. We also determine the modulation transference of
the signal and the cut-off frequency of the element itself, which is of great interest if the DOE
is going to be coupled to a transmittance system [22].
Finally we propose a direct exploration of all patterns. Modern computer capabilities
allow composing such information and obtaining a full three-dimensional view of the light
cone or even sectioning it in any plane of interest. As we will see later, this procedure
provides useful information that is not affordable by other means which can be of great
importance for proposing new DOEs configurations and applications. In this sense, three
dimensional analysis of the field is becoming an important tool for new application designs
[11,23].
2. Chromatic aberration in diffractive optical elements
Let us consider a phase-only element u0 ( r0 ) where r0 = ( x0 , y0 ) illuminated by a
monochromatic plane wave whose wavelength is λ0 . Fresnel light distribution at a distance z
is given by:
u ( r, z, λ0 ) =
exp ( i 2π z / λ )
iλ z
+∞
× ∫ u0 ( r0 )
−∞
⎛
exp ⎜ i
⎝
⎛ π
exp ⎜ i
⎝ λ0 z
ρ
π 2⎞
ρ ⎟×
λ0 z ⎠
2
0
⎞
⎛
⎞
2π
r·r0 ⎟
⎟ exp ⎜ −i
λ
z
0
⎠
⎝
⎠
(1)
dr0
where r = ( x, y ) , and where ρ = r , ρ0 = r0 .
As it is known if the object transmittance is not sensible to wavelength, changes in the
illuminating wavelength produce the same Fresnel patterns but displaced a distance z '
depending on the wavelengths ratio, i.e.:
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u ( r, z, λ ) =
exp ( i 2π z / λ )
iλ z
⎛
exp ⎜ i
⎝
+∞
⎛
× ∫ u0 ( r0 ) exp ⎜ i
⎝
−∞
=
π 2⎞
ρ ×
λ z ⎟⎠
π 2⎞
2π
⎛
⎞
ρ0 ⎟ exp ⎜ −i r ⋅ r0 ⎟dr0
λz ⎠
λz
⎝
⎠
exp ( i 2π ( λ0 / λ ) z / λ0 )
iλ0 z ( λ λ0 )
+∞
⎛
× ∫ u0 ( r0 ) exp ⎜ i
⎜
⎝
−∞
⎛λ
⎛
≡ exp ⎜ i 2π z ' ⎜
⎜
⎝
⎝
2
⎛
exp ⎜⎜ i
⎝
⎞
π
ρ 2 ⎟⎟ ×
λ0 z ( λ λ0 ) ⎠
(2)
⎞
⎛
⎞
π
2π
ρ 02 ⎟⎟ exp ⎜⎜ −i
r ⋅ r0 ⎟dr0
⎟
λ0 z ( λ λ0 ) ⎠
λ0 z ( λ λ0 )
⎝
⎠
− λ02
λ0 λ
2
⎞⎞
⎟ ⎟⎟ u
⎠⎠
( r , z ', λ0 )
being the displacement distance z ' :
z' = z
λ
λ0
(3)
In addition, the aberrated pattern at a distance z is equivalent to the original one that was
located at a distance z’. We should note that the phase factor multiplying the amplitude field
does not affect both the intensity and the inner light distribution in the patterns. Therefore its
presence can be ignored. Nevertheless, we will maintain it for the sake of completeness
Correction of chromatic aberration is usually done by introducing a dependence of the
wavelength in the object transmittance which may compensate Fresnel displacement. Since
refractive chromatic aberration has the opposite dependence of wavelength, achromatic
designs combining DOEs and lenses have been widely studied (e.g. [5]). As it is outlined
above here, we will design our diffractive element to produce focusing in the desired distance
range for a certain wavelength λ0 which will be called the "design wavelength". Illumination
with a different wavelengths will unbalance the phases in the Fresnel integral and produce
pattern displacement according to Eq. (3).
3. Chromatic displacement in axicons
Let us assume an optical element illuminated by a uniform field and let us impose that the on
axis output intensity, after passing through the element, increases with z, i.e. I out = b·z . The
element capable of such transformation is called a linear axicon [1]. If the structure of the
element is in the form that the optical power ( 1/ f ) decreases inversely with the radial
coordinate ( ρ ), the element is called direct axicon. In this case, there is no interference
between parts of the beam coming from different annular regions. This is the type of axicon
that we use in this work. Since axicons are elements with rotational symmetry it is more
convenient to have Fresnel integral in (2) expressed in cylindrical coordinates ( ρ , θ ) :
u( ρ ,θ , z) =
exp ( i 2π z / λ )
iλ z
⎛
exp ⎜ i
⎝
2π 2 ⎞ 2π Rp
ρ
u0 ( ρ0 , θ 0 ) ×
λ z ⎟⎠ ∫0 ∫0
(4)
⎡
× exp ⎢i
⎣⎢
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2π ⎛ ρ0 2 ρρ0 cos (θ − θ ') ⎞ ⎤
−
⎜
⎟ ⎥ ρ 0 d ρ 0 dθ 0
z
λ ⎝ 2z
⎠ ⎦⎥
Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17845
being R p the aperture pupil radius.
Although the axicon is defined with a linear dependence on z, usually it is not needed a on
axis constant growing energy along the optical axis, but the major part of it to be concentrated
in a focal line along the z axis from distances f 0 to f1 . In this case, the following condition
must be fulfilled:
Rp
2π
∫
I in ( ρ ') ρ ' d ρ ' =
R1
f1
∫
I out ( z ) dz
(5)
f0
where R1 is the inner aperture radius, and I in ( ρ ) represents the intensity transmitted by the
axicon in terms of the radial coordinate, and I out ( z ) represents the intensity along the axis.
With these conditions, the transmittance T ( ρ0 ) of a direct linear axicon results [1]:
⎛
2π
(1 + a ) ρ02 + f 02 − aR12
⎜
⎝
λ0
1+ a
T ( ρ0 ) = exp ⎜ −i
T ( ρ0 ) = 0
⎞
⎟
⎟
⎠
ρ0 ∈ ⎣⎡ R1 , R p ⎦⎤
(6)
elsewhere
being:
a=
f12 − f 02
,
R p2 − R12
(7)
We have chosen the direct linear axicon because it is of easier implementation in front of
other types of axicons. Detailed discussion about performance and characteristics of the linear
axicons can be found in the bibliography and will not be further discussed here.
Figure 1 shows the on-axis intensity for a linear axicon designed according to the above
Eqs. (6) and (7), for three wavelengths equivalent to the Fraunhofer spectral lines, E, C’,and
F’ which correspond to the green mercury line ( λG =546.1 nm) and to the blue and red
cadmium lines ( λB =480.0 nm, λR =643.8 λR=643.8 nm). The phase transmittance of the DOE
has been designed for the green color.
Results presented correspond to numerical calculations which have been done according
to a far distance propagation distance algorithm presented in [20,21]. Exact parameters that
have been used are summarized in Table I. All variables have been set in order that this
particular design could be implemented in a standard commercial SLM. Number of samples
used for this simulation is set in N=1024.. The closer focusing distance f0 has been selected to
be not shorter than the Nyquist focal length imposed by the size of the window. The radii R1
and Rp have been selected as the minimum and maximum circular aperture that can be drawn
in a SLM. We can see in Fig. 1 that when these elements are illuminated with polychromatic
light, axicons may produce the expected displacement of the focusing region, in accordance
with Eq. (3). Table 1 also indicates the values of the focusing ranges for each wavelength.
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Table 1. Constants used for numerical calculations of propagated fields from DOES: λG corresponds to design
wavelength. f0 and f1 are extreme distances of the focal segment, N is the number of samples used for discrete
calculations. R1 and Rp are the inner and the aperture radii of the DOE. We also show the focusing range.
DESIGN PARAMETERS
f0
λ0
546.1 nm
550 mm
FOCUSING RANGE
643.8 nm
λR
λG
λB
Rp
f1
700 mm
N
1024
R1
0.1 mm
466.5 mm
546.1 nm
f0λ0/λR
f0
550 mm
f1λ0/λR
f1
700 mm
480.0 nm
f0λ0/λB
625.7 mm
f1λ0/λR
796.3 mm
6.95 mm
593.7 mm
Fig. 1. Numerical simulation of on axis irradiance for a linear axicon illuminated with red,
green and blue light.. The DOE has been designed for focusing in the region [f0=550 mm,
f1=700mm] when illuminated with λG wavelength.
4. Achromatic axicons
As we can see in Fig. 1, when illuminated with polychromatic light, linear axicons produce
colored segments displaced along the optical axis. Notice that if we extend the upper limit of
the focusing line, we can achieve superposition of the three bands of color and obtain a colorbalanced region between red and green and green and blue colors and even an achromatic
region. The condition that must be fulfilled for obtaining an achromatic focusing region is that
the two extreme colors overlap. Thus, the focal depth produced by the element must be larger
than the displacement produced by the chromatic aberration. According to Eq. (3),
illumination with λR and λB will displace the region to the left and to the right to
λG
λ
λ
λ
, f1 G ] and [ f 0 G , f1 G ] respectively. Superposition between blue and red regions
λR
λR
λB
λB
will happen only while:
[ f0
f1
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λG
λ
> f0 G
λR
λB
→
f1 λR
>
f0 λB
(8)
Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17847
Therefore, by appropriately selecting the values of f0 and f1 it is possible to obtain an
achromatic focusing region, where all considered wavelengths are being focused. In Fig. 2,
we show the effect just explained. We have selected a new range [ f 0 , f1 '] with now being
f1 ' = 940 mm fulfilling condition in Eq. (8) . Numerical calculation depicted in Fig. 2 show
that superposition of different wavelengths produces a region where overlapping of all colors
is accomplished, and then achromatism is expectable.
Fig. 2. Numerical simulation of on axis irradiance for a linear axicon illuminated with red,
green and blue light. The DOE has been designed for green light focusing in an extended
region [f0=550 mm, f1’=940mm] when illuminated with λG wavelength. Notice that
superposition of colors is obtained from 650 mm to 780 mm.
Notice that in both figures axial distribution is rough. This can be corrected by introducing
an apodizing filter in order to soften the axicon responses [17]. However we have preferred to
present the distribution as it is and no performing additional transformations, since it does not
affect to the chromatic effect we are analyzing. Notice also that the total energy peaks have
diminished with respect to those represented in Fig. 1. This is a direct consequence of
expanding the depth of focus of the DOE.
In order to analyze the optical quality of the achromatic region, in Fig. 3 we present the
radial MTF of the light distribution in the region of interest, corresponding to an incident
plane wave, for the three colors, λR , λG and λB . Frequency values have been normalized to
the cut off value of a normal lens with the same aperture, illuminated with λG and focusing in
the center of the interval (745 mm) In order to better visualize the focalization we have only
represented the region up to 20% of the maximum. Values over this factor are saturated.
Notice that the MTFs widen at the region where focalization is produced. We can also
observe that frequency response of the element is not uniform along the focal segment, but it
forms a lobe. The center of the lobe is located at the distance where we achieve the maximum
MTF value. This distance coincides with the center of the focusing segment for each
wavelength. We observe that there is a region where the MTF lobes for the three colors
coincide. Although there is no coincidence between the maxima, we can define a band where
the MTF values are balanced.
In Fig. 4 we depict the MTF for distances at the beginning, middle and end of the
superposition band for the three considered wavelengths. Results are low, compared with a
standard lens, but we have to consider that we are representing an achromatic depth of focus
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of 120 mm. The result obtained here is similar to that presented in [7] by multiplexing several
lenses. However, our design differs in that we obtain an achromatic focus. In addition a slow
decay along the axis is observed, instead of clear peak. Since we are extending the effect of an
axicon, instead of multiplexing several elements, off-axis energy is relatively low, and a more
efficient element is accomplished.
Fig. 3. Radial MTF of the linear axicon. Notice widening of the function on the region of light focalization.
Fig. 4. Radial MTF for a linear axicon at the beginning (618 mm), middle (673 mm) and end
(732 mm) of the region of interest. Notice that balanced MTF is obtained for three colors.
In Fig. 5, we show an animation with the RGB simulation of the 3D structure of the beam
as it propagates around the region of interest. The figure has been generated by calculating the
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propagated light patterns for each wavelength with the algorithm proposed in [20,21]. We
have assigned the different chromatic peaks to the color RGB channels, and we have merged
all the information of a plane in a single frame. Three dimensional stacking has been made
with ImageJ software [26]. In order to improve the quality of the figure, we have saturated the
energy at 1/3 of the absolute maximum of the obtained intensity. Even with this saturation
notice that off-axis energy is relatively low. There one can also observe the structure of the
achromatic region together with the other colored segments. We can see that the achromatic
region is long and narrow enough to call this element an "achromatic axicon". Notice also,
that if we are interested only in energy peaks, and not in chromatic components, the original
extension of the axicon focal line is "chromatically" extended to a new region
[ f 0 λG / λR , f1' λG / λB ] .
Fig. 5. Animation with the three dimensional structure of the axicon focusing segment. Notice
the presence of the achromatic segment.
5. Chromatic aberration in light sword elements
In [2, 12] we find other interesting elements that produce extended depth of focus with
reasonable MTF extent, which allow better contrast, but less resolution than axicons. These
elements have been called Light Sword Elements (LSOE) and are based in an angular
multiplexing of the convergence information and produce a comatic spot that rotates around
the z-axis. In this section we apply the same chromatic analysis than in the previous case to
LSOE. The transmittance function of a simple LSOE can be defined as:
T ( ρ0 , θ 0 )
⎛
⎜
= exp ⎜ −i
⎜
⎜
⎝
T0 ( ρ0 ,θ 0 ) = 0
2π
ρ0 2
λ0 f + f − f θ 0
( 1 0)
0
2π
⎞
⎟
⎟
⎟
⎟
⎠
ρ0 ∈ ⎡⎣ R1 , R p ⎤⎦ , θ 0 ∈ [0, 2π ]
(9)
elsewhere
Notice that this transmittance is equivalent to a lens but assuming that the focal distance
depends on the angle θ , i. e., for ρ0 in the pupil region:
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⎛
T ( ρ0 , θ 0 ) = exp ⎜⎜ −i
⎝
ρ02 ⎞
⎟
λ0 f (θ 0 ) ⎟⎠
2π
θ
f (θ 0 ) = f 0 + ( f1 − f0 ) 0
2π
,
(10)
Therefore, such element provides convergence of the beam in the region between f0 and f1.
However, the azimuthal dependence of the phase transmittance of these elements produces
off-axis shifts of the focal points. Therefore, the study of the radial MTF results is even more
interesting than axial irradiance, which does not provide relevant information.
The LSOE will also be affected by chromatic aberration that may displace the peaks with
respect to the design position. Again as we did in the axicon case, we impose superposition of
the two extreme peaks so, from now on, we will consider the extended interval [ f0 = 550 ,
f1 ' = 940 mm]. Under these conditions -polychromatic illumination and overlapping of the
colored focusing zones- the particular structure of the LSOE produces interesting effects. In
particular, due to the azimuthal dependence of the transmittance, the LSOE has not only
lateral and longitudinal chromatic aberration, but also shows what we call "angular chromatic
aberration". Consider that one beam converges, after passing through the LSOE at a distance:
z1 = f 0 + ( f1 '− f 0 )
θ0
2π
.
(11)
According to Eqs. (2) and (3), illumination with a wavelength λ different from the from the
design one λ0 , will produce an axial displacement of the field. Thus we will find a colored
replica of the original field at a distance:
z '1 =
λ0 ⎛
θ0 ⎞
.
⎜ f 0 + ( f1 '− f 0 )
2π ⎟⎠
λ⎝
(12)
If this displaced distance is within the range of focusing distances of the original LSOE
[ f 0 , f1 '] we can advance or delay the angular coordinate θ 0 in order to move the present
distribution to the same plane where original convergence was produced. The azimutal
coordinate θ ' that provides focusing at the same plane is given by:
λ0 ⎛
θ ⎞ ⎛
θ'
f 0 + ( f '1 − f 0 ) 0 ⎟ = ⎜ f 0 + ( f '1 − f 0 )
λ ⎜⎝
2π ⎠ ⎝
2π
⎞
⎟
⎠
(13)
And, from here we can state that illumination with two wavelengths produces, at the same
plane, two different foci with an angular relative displacement between them in the form:
θ'=
⎞
f0 ⎛ λ
λ
− 1⎟ + θ 0
⎜
f1 '− f 0 ⎝ λ0 ⎠ λ0
(14)
That is what we called the "angular chromatic aberration". This effect can be better
observed in Figs. 6(a) and 6(b), where an animation of the calculated light field as it
propagates from 550 to 940 mm and the whole structure of the beam can be observed.
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Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17851
(a)
(b)
Fig. 6. Numerical simulation of on axis (a) and three dimensional structure (b) of the LSOE
focusing segment. Notice the presence of three separated colored spots at a single plane.
Notice that no axial coincidence of the colors occurs at any plane. As in the previous case,
the energy distributions for each color have been saturated in order to improve visualization.
Unfortunately, merging the three channels causes that axial irradiance seems to be zero,
although it is not.
We can also observe that it is possible to obtain three different peaks of different color
simultaneously in the same plane. This fact could be used to split three color channels with
one single element and without additional chromatic filter.
The quality of the three peaks can be tested by obtaining the MTF of each color
separately. In Fig. 7 we show the distribution of the radial MTF along the propagation
distance. We can see that the focusing region appears clearly and there is a region where
superposition of the MTFs is observable. In Fig. 8 we show the MTFs at three different
distances in the superposition region. We see that the MTFs are comparable for the three
colors in the middle and end of the focusing region. Good balance in the entire observable
region occurs only in green and red colors. It is in this region where it is possible to make
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Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17852
color demultiplexing of a signal. Coincidence of MTFs means that using the three colored
spots for signal processing is achievable without distorting any channel with respect to other.
In [27], the authors present imaging properties of LSOEs, with similar MTFs that the ones
calculated here. Thus, it is possible to obtain simultaneously three different chromatic
channels that can be processed together or separately.
Fig. 7. Radial MTF of the linear axicon. Notice widening of the function on the region of light focalization
Fig. 8. Radial MTF for a LSOE at the beginning, middle and end of the region of interest.
6. Conclusions
We have analyzed the performance of two different DOEs with extended depth of focus under
polychromatic illumination. We have first analyzed a linear axicon. We have showed that,
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Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17853
using their particular characteristics, we can extend the focusing region and force overlapping
between the different chromatic segments. Superposition of the three colors λR , λG and λB
allows obtaining an achromatic focusing region. Although the resulting element does not have
a very high MTF, it may have application in designing new achromatic implants for the
human eye, whose standard cutoff frequency is 0.18 times the single lens, or equivalently 30
cycles per degree [28].
We have also studied the performance of the light sword optical element (LSOE) under
polychromatic illumination. It presents what we called the angular chromatic aberration,
which consists in angular displacements of the same focusing spots for different wavelengths.
Its particular configuration does not permit obtaining an achromatic focusing region, but
allows obtaining three colored focused spots at a single plane. The MTFs analysis shows that
quality of the peaks is good enough to allow their use in color signal processing, since it
allows chromatic channel splitting with a single element.
Acknowledgments
This work has been financed by the Spanish Ministerio de Educación y Ciencia through the
projects nr. FIS2005-05053 and FIS2006-13037-C02-02.
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Received 29 Aug 2007; revised 17 Oct 2007; accepted 26 Oct 2007; published 13 Dec 2007
24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17854